Problem 39 - Precalculus with Limits

Question

Using Heron's Area Formula use Heron's Area Formula to find the area of the triangle. $$ a=2.5, \quad b=10.2, \quad c=9 $$

Step-by-Step Solution

Answer

After performing the computations, the final area, using Heron's formula, is 7.96 square units.

Step 1: Calculate the Semi-Perimeter

First, the semi-perimeter of the triangle needs to be calculated using the formula mentioned above: \( s = (a + b + c)/2 \). Therefore, \( s = (2.5 + 10.2 + 9)/2 = 10.85

Step 2: Apply Heron's Formula

Now, apply Heron's formula to find the area of the triangle using the calculated semi-perimeter and the given sides: \( A = sqrt{[s(s - a)(s - b)(s - c)]} \). So, \( A = sqrt{[10.85*(10.85 - 2.5)*(10.85 - 10.2)*(10.85 - 9)]} \).

Step 3: Calculate the Area

Evaluate the expression found in the previous step to find the area of the triangle: \( A = sqrt{[10.85*(8.35)*(0.65)*(1.85)]} \). After carrying out the calculations, the final value for the area will be found.

Key Concepts

Semi-Perimeter of a Triangle

The semi-perimeter of a triangle is a key part of calculating the area using Heron's formula. It is denoted by the letter \( s \) and represents half of the triangle's perimeter. Calculating the semi-perimeter is straightforward. You simply add up all three sides of the triangle—let's call them \( a \), \( b \), and \( c \)—and then divide that sum by two. The formula looks like this:

\[ s = \frac{a + b + c}{2} \]

This concept helps break the problem down into simpler terms, making it easier for you to find the triangle's area. For our exercise, we plug in the given side lengths \( a=2.5 \), \( b=10.2 \), and \( c=9 \) into the formula to find \( s \):

\[ s = \frac{2.5 + 10.2 + 9}{2} = 10.85 \]

Knowing the semi-perimeter is crucial because it acts as the foundational step in our area calculation.

Triangle Area Calculation

Once the semi-perimeter is calculated, finding the area of the triangle becomes an application of Heron's formula. Heron's formula is an elegant way to calculate area without needing to know the height of the triangle. It's particularly useful for irregular triangles where traditional base-height methods may not be straightforward.

Here is Heron's formula for triangle area:

\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]

In this formula:

• \( s \) is the semi-perimeter we just calculated
• \( a \), \( b \), and \( c \) are the sides of the triangle

By substituting \( s = 10.85 \), \( a = 2.5 \), \( b = 10.2 \), and \( c = 9 \) into this formula, we start the process:

\[ A = \sqrt{10.85 \times (10.85 - 2.5) \times (10.85 - 10.2) \times (10.85 - 9)} \]

This simplification will give the correct calculation for the area.

Geometry Formulas

Understanding and navigating geometry often involves a variety of formulas, like Heron's formula. This foundational mathematical discipline uses these formulas to help describe and quantify shapes, sizes, and other properties of figures and spaces.

For triangles, besides Heron's area formula, you might regularly encounter calculations involving:

• Pythagorean theorem for right-angled triangles
• Trigonometric formulas involving sine, cosine, and tangent
• Basic area formulas like \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \)

These formulas serve as tools to explore the properties of triangles more deeply. For example, Heron's formula is particularly useful when you have the lengths of all sides but no height. Having a toolkit of these geometric formulas can make solving math problems more efficient and less daunting.